Least Common Multiple Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Least Common Multiple.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The smallest positive integer that is divisible by each of two or more given numbers—where their multiples first coincide.

The first number that appears in both times tables—where two counting sequences land on the same value.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The LCM is the first value that appears in both numbers' multiple lists.

Common stuck point: The procedure for least common multiple is the easy part; the trap is picking the GCF by mistake. Asking "Am I looking for the smallest number that every given value divides into evenly?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I looking for the smallest number that every given value divides into evenly?

Worked Examples

Example 1

easy

Find the LCM of

12

and

18

Answer

36

First step

Prime-factor each number:

12 = 2^2 \times 3

and

18 = 2 \times 3^2

Full solution

2
For the LCM, keep the highest exponent of each prime that appears: $2^2$ and $3^2$ .
3
Multiply those factors: $2^2 \times 3^2 = 4 \times 9 = 36$ , so the LCM is $36$ .

The LCM is the product of all prime factors, each raised to the highest power from either number. The LCM is the smallest number that both original numbers divide into evenly.

Example 2

medium

Find the LCM of

8

12

, and

15

Example 3

easy

Find the LCM of

4

and

10

by listing multiples.

Example 4

medium

Use prime factorization to find the LCM of

24

and

36

Example 5

medium

Find the LCM of

6

10

, and

15

Example 6

medium

A jogger runs a loop every

4

minutes; a cyclist completes it every

6

minutes. If they start together, after how many minutes will they next be together at the start?

Example 7

hard

Three traffic signals blink every

9

12

, and

15

seconds. If they blink together at time

0

, how often after that do all three blink together?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the LCM of

9

and

15

Example 2

easy

Two lights flash every

6

seconds and every

8

seconds. If they flash together now, after how many seconds will they flash together again?

Example 3

easy

Find the LCM of

4

and

6

Example 4

easy

Find the LCM of

3

and

5

Example 5

easy

Find the LCM of

2

and

8

Example 6

easy

Find the LCM of

6

and

9

Example 7

easy

Find the LCM of

5

and

10

Example 8

easy

Find the LCM of

4

and

5

Example 9

easy

What is the LCM of any number

n

and itself?

Example 10

easy

Find the LCM of

6

and

8

Example 11

medium

Use prime factorization to find the LCM of

12

and

18

Example 12

medium

Find the LCM of

8

and

20

using prime factorization.

Example 13

medium

Find the LCM of

4

6

, and

10

Example 14

medium

Add

\frac{1}{4}+\frac{1}{6}

using the LCM as common denominator.

Example 15

medium

Two bells ring every 12 and 18 minutes. After how many minutes do they ring together?

Example 16

medium

When does multiplying the two numbers correctly give their LCM? Illustrate.

Example 17

medium

Find the LCM of

9

and

15

Example 18

challenge

Given

\gcd(a,b)=4

and

a\times b=192

, find

\mathrm{lcm}(a,b)

Example 19

challenge

Find the smallest number that leaves remainder 0 when divided by 6, 8, and 9.

Example 20

challenge

Two gears have 14 and 21 teeth. How many rotations of the smaller gear until both return to start together?

Example 21

medium

Find the LCM of

10

and

15

using prime factorization.

Example 22

medium

Add

\frac{1}{6}+\frac{1}{9}

using the LCM denominator.

Example 23

easy

Find the LCM of

7

and

14

Example 24

easy

Find the LCM of

3

and

7

Example 25

easy

Find the LCM of

8

and

12

Example 26

easy

Find the LCM of

5

and

7

Example 27

medium

Find the LCM of

14

and

21

Example 28

medium

Find the LCM of

15

and

20

Example 29

medium

Add

\dfrac{1}{8} + \dfrac{1}{12}

using the LCM denominator.

Example 30

medium

Three buses leave the station every

15

20

, and

30

minutes respectively. If they leave together at noon, when do they next all leave together?

Example 31

medium

Find the LCM of

16

and

24

Example 32

medium

Find the LCM of

25

and

35

Example 33

medium

Subtract:

\dfrac{5}{6} - \dfrac{3}{8}

using LCM as denominator.

Example 34

medium

Find the LCM of

18

and

30

Example 35

hard

Find the smallest positive integer divisible by

4

5

6

, and

9

Example 36

hard

\gcd(a, b) = 12

and

\text{LCM}(a, b) = 180

. If

a = 36

, find

b

Example 37

hard

What is the smallest integer greater than

1

that, when divided by

2, 3, 4, 5,

6

, leaves remainder

1

Example 38

hard

Find the LCM of

2^3 \cdot 3^2 \cdot 5

and

2^2 \cdot 3^3 \cdot 7

Example 39

challenge

Find the smallest positive integer

n

such that

n

is a multiple of

7

and

\text{LCM}(n, 12) = 84

Example 40

challenge

a

and

b

are positive integers with

\text{LCM}(a,b) = 36

and

\gcd(a,b) = 6

. List all possible unordered pairs

\{a,b\}

← Back to Least Common Multiple Formula Explained Practice Mode

Related Concepts

Multiples Adding Fractions

Background Knowledge

These ideas may be useful before you work through the harder examples.

multiples